A New Equation of Mean Velocity in Flow, Its Verification and
Consequences: H.B. Fischer's Experiment in the Caltech Canal.
ALFREDO JOSE CONSTAIN ARAGON
Fluvia Tech.
Bogotá, Calle 132A # 19-64 (301-2)
COLOMBIA
Abstract: - The Chezy-Manning equation has long been the macroscopic description of flow in real channels, and
its foundation is the balance of Newtonian mechanical forces. This Article presents the development and
verification of a new equation of average velocity in the flow, defined this time in terms of electric forces (Van
der Waals), having the same structure as the classical equation. Its application can be very broad, since in addition
to having a new and powerful theoretical tool, in engineering practice it allows reducing the degree of uncertainty
in the measurement of natural channels. The validity of the Elder equation, which links hydraulics and Dispersion,
is analyzed here and an adjustment to its application is proposed, considering the Longitudinal Transport
Coefficient as a function of time.
Key-Words: - Hydraulics; Fick, Chezy-Manning; Elder; turbulence, state functions.
1. Introduction
Environmental and Hydraulic Impact studies use
advanced software models that require feeding with
precise data that adequately represents the complex
reality of natural channels. Although these models
have advanced a lot, serious limitations persist in the
quality and quantity of the data series, since the
theoretical tools have not advanced at that pace, and
isolated techniques and concepts are available, which
are approximate in their domain. local” application,
its general validity is limited and meager, especially
in large channels. The foundations of the problem
and an alternative model that can solve it are then
presented.
2. Problem Formulation.
2.1 The problem of modeling environmental
impacts: The basic concepts.
Industrial Society generates an astronomical amount
of waste of all types that in a certain percentage end
up in natural channels, negatively impacting the
quality of the resource. To characterize and control
this, multiple efforts have been developed in the
physics and chemistry of the evolution of pollutant
transport in flows.
A first result was Taylor's one-dimensional mass
balance equation in 1954, where C and U are average
values of the concentration and velocity over the
cross section of the flow, Ayz, and D as the
longitudinal dispersion coefficient. [1]:

 
 󰇡
󰇢
 (1)
Although it was accepted that the main cause of the
dispersion of the pollutant was the variation of the
elemental velocities over Ayz, thanks to the shear
effect, the process could be approximated by a
Fickian expression (including concentration
gradients). Likewise, Taylor found that a value of D
for the case of a long, straight tube, with Radius a, the
density of water, ρ, and τo, as the “shear stress”
(tangential tension of the edge in the viscous
medium).

(2)
Based on this analysis, in 1959, J.W. Elder[2],
applying velocity variations on the vertical axis, Δvz,
in accordance with the logarithmic (turbulent)
definition of the velocity distribution in a viscous
medium, according to Prandtl. [3][4]. Figure 1.
Received: March 26, 2024. Revised: August 19, 2024. Accepted: September 23, 2024. Published: October 17, 2024.
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Figura 1.- Perfil no lineal de velocidad
Elder found for an open channel of infinite width, that
D could be expressed as:
 (3)
Where h is the average depth of the flow, S its slope,
and g the acceleration due to gravity.
Although Elder's definition could be applied to
natural flows with some approximation, the
experimental values in natural channels were
dispersed in a ratio of 1 to 15, calling into question
even the Fickian concept of mass transport in
turbulence. Furthermore, if we consider that the
analytical solution of the differential equation (1) is
the basic Fick expression, with M as the mass
suddenly poured into the flow, and Xo the
observation distance from the injection point:
󰇛󰇜
󰇛󰇜
 (4)
Despite the rigor of Taylor and Elder's previous
arguments, it was verified that not only did the D
values not correspond to those actually observed in
natural channels, but that the asymmetric shape of the
measured Fickian curve had an “abnormal” bias.
Figure 2.
Figure 2.- Sesgo anormal en la respuesta real.
To help solve this incongruence between theory and
reality, many hypotheses and procedures were
presented, with varying degrees of success (and
failure), among them the method of H.B. Fischer
[5][6] who considered, unlike Elder, that the primary
cause of turbulent diffusion was not the vertical
distribution of velocities, v(z), but the transverse
distribution of velocities, v(y).
2.2 Vertical and transversal
velocity distributions: Two sides of the same
reality.
A first consideration to solve the problem is to ask a
simple first question: How different are the turbulent
velocity distributions in the vertical axis (Elder) and
those in the transverse axis (Fischer)? Or put another
way: Is their nature really different?
To resolve these questions, it must be observed that
the basic mechanism for generating the “dispersion”
itself is the “random separation” of pairs (contiguous)
of fluid particles, due to a “fluctuation” (difference)
in the velocities in that pair. of very close points, and
that produces the separation at different distances of
the two particles that were initially “united” (at the
same point). That is, how the action of a du generates
a dX. [7] Figure 3.
Figure 3.- Random separation of a pair of particles
If we idealize the longitudinal velocity field in the
cross section of the flow, we can separate an average
velocity, <U>, and the alterations (pulsations) that
are an “indivisible” mixture between wave and
turbulent motion, du, [8][9]
󰇛󰇜  (5)
If it is considered that “the entire” systematic part of
the water movement, and therefore equal, is
concentrated in <U>, then the total variations will be
concentrated in du.
Since the part of this definition that generates the
separation is du, the question now arises whether this
differential will be different if it is placed on the Z
axis or the Y axis. Clearly not, because the speed
differentials are due only to turbulence, and if this
is considered approximately homogeneous and
isotropic [10], we have approximately that:
   (6)
Therefore, the hypothesis of H.B. Fischer, an
essential difference in the distribution of increases in
turbulent velocity according to the axis, as a cause of
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the theory-reality discrepancies for the movement of
tracers, is not supported as an underlying reason.
2.3 The Longitudinal Diffusion Coefficient as a
function of time.
Now, although in the various developments proposed
to solve the problem, the value of the Coefficient D
itself is a function of certain parameters, as seen in
equations (2) and (3), if these do not vary, it is
assumed that D would not vary either. , that is, it is
not considered a function of time, since this fact
would introduce certain problems of interpretation of
the dispersion mechanism, such as the idealization of
the “uniform flow” condition [11][12]. An interesting
idea that has been presented previously, but has not
been developed in detail [13] is that the Coefficient
D should be a function of time.
If you see closely, the temporal nature of the
Transport Coefficient is a no small issue from the
point of view of Mechanics, it involves the notion of
relative motion and its relationship with various
inertial observers [14].
A first mobile observer located at the peak of the
distribution, moving at speed U, does not compose it
(U=0) and describes the Fickian dynamics of a
symmetrical tracer plume, as shown in Figure 4.
Figure 4.- Symmetrical kinematic composition of the
plume.
A second fixed observer located on the bank of the
channel does compose the velocity U, such that U>0,
and describes the Fickian dynamics of an asymmetric
tracer plume, as shown in Figure 5
Figure 5.- Asymmetric kinematic composition of the
plume.
In this case, with the average velocity U, two other
opposite velocities are composed: +vd and -vd,
associated with the movement of diffusion,
expanding to both sides of the central point of the
mass injection. This composition leads to the initial
segment of the Fickian curve being much steeper than
the subsequent segment, generating the bias that is
observed experimentally.
The net velocity, composed kinematically as a
function of time, can then be described:
󰇛󰇜   (7)
Since the motion of the tracer is described as a
Brownian one, it can be written in a general way as
[15]:



(8)
Here, the “characteristic time”, τ, can be expressed as
a function of the general time, t, in the following way,
incorporating the Feigenbaum Number, δ, in such a
way that the self-similar, chaotic nature of the
phenomenon is manifested. [16]
  (9)
And you can define a State Function, Φ(U,E,t),
that complies with the following:
 (10)
And


(11)
From here then, the average velocity is:

(12)
This equation has the same mathematical structure as
the classical Chezy-Manning relationship, for the
mean velocity of uniform flow, using mechanical
quantities. [17]:
(13)
In equation (13) the source magnitude of the
movement is gravitation acting through the slope, S≈
dy/dx, while in equation (12) they are the electric
forces (Van der Waals) acting through the state
function , Φ.
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This function reflects the exchange of energy
between the mass of the tracer and its liquid
environment, basically due to the expenditure of
Enthalpy of formation, which is expelled as heat to
the outside. As this process depends on the square
root of the Concentration, √C, it can be shown that
the expression of Φ depends on this factor.[18]
Figure 6.
Figure 6.- State function Φ(t).
Actually, what the Φ(t) curve represents is a phase
transition, in which the solid tracer becomes a gas,
which occurs at the point Φ≈0.38, at which almost all
of the mass of the tracer is lost. their mutual
interactions, and the description from there is
properly of water in turbulence. This point is very
important because the practical description of this
state does not require the use of non-linear
differential expressions.[19]
2.1 Data obtained from the tracer curve.
When a mass M of tracer is injected abruptly, a tracer
concentration distribution is generated that
corresponds to the solution of the Taylor equation
(1), and is the Fick equation (4), only if we do not
have consider that the Coefficient of Dispersion is a
function of time, D(t), its development will not
appropriately replicate the experimental data, in
particular it will present incorrect bias.
The way to correct this problem is to solve for D(t)
from equation (12) and replace it in (4), as shown
below, where to is the time measured after passing
the peak of the distribution at the point of
measurement:

(14)
And then:
󰇛󰇜
󰇛󰇜
󰇛󰇜 (15)
In this expression Q≈U*Ayz, is the flow rate (in l/s)
and t is the general time. The validity of this equation
is verified when the experimental curve agrees with
the model, and the correct dispersive data are those
that allow this coincidence: basically, the Φ(to) data,
the flow rate and the average velocity, U(to), having
the Xo data and the Mass, M in μg if a fluorescent
tracer is used, such as Rhodamine WT or Fluorescein.
These data are obtained from the special
measurement equipment, FLUVIA F-1[20], but can
also be calculated from the tracer curves of cases that
have not been measured with said equipment. Figure
7.
Figure 7.- Tracer measurement equipment.
2.3 Calculation of roughness and its
verification with Elder's equation.
The existence of a definition of the mean flow
velocity that is based on Newton forces, the Chezy-
Manning equation (13), and also on Coulomb forces
(12), allows the calculation and verification of
hydraulic data (sometimes difficult to obtain) from
Dispersion data (easier to achieve). In this way, a data
such as the Roughness of a flow, n, can be stated by
equating (12) and (13) as follows:


(16)
3 Analysis of an experimental case.
To apply the methodology explained here, an
experiment carried out by H.B. Fischer in the Caltech
calibrated channel, in 1966. Figure7. [21][22]
Figure 7.- Caltech Calibrated Channel.
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Likewise, the two consecutive results of injecting a
saline solution are shown, at Xo1=14.05 m and
Xo2=25.06 m. [23] Figure 8
Figure 8.- Salt tracer curves at two distances.
3.1 Channel data.
This experiment was carried out in the 40.0 m
calibrated channel of the M. Keck Laboratory of
hydraulics of CALTECH, especially the experiment
called “Series 2700”, with adjustable slope and bed
material. The geometric data of the channel are as
follows, Table 1:
Table 1: Channel data
Specifications
Characteristics
Metrics.
Shape
Rectangular
Ayz≈0.140 m2
Distance Xo1
linear
Xo1=14.06 m
Distance Xo2
linear
Xo2=26.06 m
Bottom
Smooth
Wide=1.09 m
Sides
Smooth
Depth=0.128 m
Slope
Mechanical adj.
S≈0.0002 min.
Hydraulic radius
Mean value of flow
R≈0.104 m
3.2 Datos de la Advección y dispersion y
datos de la modelación con la ecuación (4).
Los datos del transporte advectivo y dispersivo se
muestran en el siguiente Table 2.
Table 2: Tracer experiment two points data
Specifications
Metrics.
Mass of saline
solution (NaCl).
M≈40500 mgr
Time of observation
at Xo1
to1≈ 38.5 s
Time of observation
at Xo2
to2≈ 68.8 s
Mean Velocity
U≈0.372 m/s
Shear Velocity
U*≈0.0159 m/s
Discharge
Q≈ 50.8 L/s
State function at to1
Φ1≈0.137
State function at to2
Φ2≈0.130
Longitudinal
Diffusion
Coefficient at to1
D1≈0.0106 m2/s
Longitudinal
Diffusion
Coefficient at to2
D2≈0.0169 m2/s
The modeling of the two tracer curves using the
“Modified Fick” equation (4) is shown below,
superimposed on the experimental data, showing that
the simulation is quite good, representing the
asymmetries of the real curves. Figure 9.
Figure 9.- Modeling of saline tracer curves at two
distances.
3.3 Calculation of the Manning Roughness of
the channel from the new equation.
The great advantage is that this experiment with a salt
tracer is profusely documented, although the value of
the channel roughness is not in the list of values, so it
is interesting to calculate it with (16) and then review
its probable value in a Table of roughness.
Considering the initial data from both the channel and
the tracer experiment, its value is solved according to
equation (16). For the two measurement points, like
this:
 

 


  (17)
And:
 

 


  (18)
The two values are coincident with the figure of
n≈0.0085, a value that is the reference figure for
artificial laboratory channels, in the text of V.T.
Chow [24].
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3.4 Verification of the Slope, S, with Elder's
equation.
Considering that Elder's equation is valid for all types
of velocity distribution (vertical or transversal), it
will be applied to this experiment for which the
maximum depth, h≈0.128 m, and the shear velocity,
U*≈0.0159 m/s, are known, therefore, for the two
distances the Slope is solved:
 (19)
Then:
 
 
 (20)
And:
 
 
  (21)
The average value of these values is <S>≈0.00028,
that is, with a percentage error of 40% but within the
same order of magnitude. This error probably
corresponds to the fact that, as Elder's formula for
“D” is defined, this parameter does not depend on
time, but as has been shown here, this dependence
does exist, therefore the behavior of the Elder relation
actually it should be expressed graphically as in
Figure 9.
Figure 9. Time dependence of the Elder equation.
This means that Elder's equation must necessarily be
interpreted as a function of time, and that there will
be an “optimal observation time”, too. This time can
be established with an “Estimation Function”, as
expressed below.
󰇡
󰇢󰇡
󰇢
 (22)
Here for simplicity we have written “C” is the Chezy
constant, which is:
 (23)
This estimate is exactly F=5.93, for the optimal time,
t=too. To determine this optimal time, interpolations
of values of the time-dependent factor Φ2*τ are made
(since the other values are not considered temporal
functions with known values) and we have that
F≈5.95 (with an error of 0.4%) with Φ ≈0.132 , too≈
46.4 s. and D(too)≈ 0.0121 m2/s. for S≈0.0002, in the
Caltech channel experiment.
3.5 Methodology to establish approximate
values of hydraulics based on the average
velocity equations and the Elder equation.
With the tracer, Dispersion and Advection data are
obtained, which can be approximately extrapolated
from the development of the State Function. In this
way, values can be proposed at a relatively large
distance from Φ(t). Using an average value of the
depth, h, and the average width, which are measured
by bathymetry or by detailed observation, the slope
of the channel, S, is obtained. , the Cross Section,
Ayz, and from there we can solve for the Manning
roughness, n. Figure 10.
Figure 10. Approximate measurement of “far”
values of hydraulics and geomorphology by Φ(t).
This extrapolation is possible as long as the channel
is in “dynamic equilibrium”, when the production of
entropy is minimal and the differences between the
values have minimum variance.[25].
4. Conclusions
1.- With a new equation for the average velocity of
the flow in a non-uniform regime, it is possible to
introduce a Longitudinal Transport Coefficient, D(t),
as a function of time, which eliminates the errors that
are introduced in the formation of the tracer plume
model. allowing the experimental bias to be
appropriately reproduced.
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2.- This temporal dependence of the longitudinal
transport coefficient arises from the existence of a
State Function that describes the evolution of the
tracer in turbulence. This function describes the
phase transition of the tracer in the flow, and its
statistical coupling with it.
3.- Historical objections to the validity of the Elder
equation, which defines D(t) are not founded, since
turbulence can be approximated as an isotropic
phenomenon, and the nature of the vertical
distribution of velocities is not distinguished in
principle. of that of Lateral Distribution.
4.- Using the Chezy-Manning (13), Van der Waals
(12), and Elder (3) equations with adjustments, it is
possible to propose a methodology for measuring
parameters in natural channels, minimizing
uncertainties.
5.- This methodology is verified in the experiment
carried out by H.B: Fischer in the Caltech calibrated
channel in 1966, which was documented in great
detail.
References
[1] Fischer H.B. Dispersion predictions in natural
Streams. Journal of Sanitary Engineering. October
(1968).
[2] Elder J:W: The dispersion of market fluid in
turbulent shear flow. Journal of fluid mechanics. 5.
Part 4. May (1959)
[3] French. R. Hydraulics of open channel. McGraw-
Hill, (1985)
[4] Nekrasov B. Hidraulica. Editorial Mir, Moscú.
(1968).
[5] Fischer H.B. PhD Thesis. (1966)
[6] Fischer H.B. The mechanics of dispersion in
natural streams. Journal of Hydraulics Division. HY
6. November (1967).
[7]Constaín A. Dispersión Random Walk,
irreversibilidad y velocidad en flujo no uniforme.
Revista Guillermo de Okham. Cali, (2005).
[8]Simonenko S.V. Non-equilibrium statistical
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[9] Meyer R.E. Introduction to mathematical fluid
dynamics. Dover, New York. (1971)
[10] Simonenko S.V. Ibid (1971)
[11] Holley E.R. Unified view of diffusion and
dispersion in natural streams. Journal of the
Hydraulic division. ASCE, March (1969).
[12] French R. Ibid (1985).
[13]Cushman Roisin B. Beyond eddy diffusivity: An
alternative model for turbulent dispersion. Environ.
Fluid Mech. (2008)
[14] Frish S. & Timoreva A. Curso de física general.
Tomo 1. Editorial Mir, Moscú, (1969).
[15] Einstein A. Investigations on the theory of the
Brownian movement. Dover. New York.(1956).
[16] Constaín A., Peña G. & Peña C. Función de
estado de evolución de trazadores, Ф(U,E,t), aplicada
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la turbulencia. Revista Ingeniería Civil, CEDEX,
Madrid. (2022).
[17] Vennard J. Elementos de la mecánica de fluidos.
C.E.C.S.A. México.(1965).
[18] Constaín A. The Svedberg number, 1.54, as the
basis of a State function describing the evolution of
turbulence and dispersion. Chapter Intech Open
book, London. To be published next.(2024).
[19] Anderson P.W. More is different.
Science,177(4047). (1972).
[20] Constaín A., Lemos R. & Carvajal A.
Tecnología IMHE: Nuevos desarrollos de la
hidráulica. Revista Ingeniería Civil, CEDEX,
Madrid. No. 129. (2003).
[21] Constaín A. Aplicación de una ecuación de
velocidad media en régimen no uniforme : Análisis
detallado del transporte en el Canal Caltech usando
Excel. Revista Ingeniería Civil, CEDEX, No. 170.
(2013).
[22] Constaín A. Revalidación de la ecuación de
Elder para la medición precisa de Coeficientes de
dispersión en flujos naturales. Revista DYNA,
Medellín, No.81. (2014).
[23] Constaín A. & Corredor J. Ecuación de Elder:
Una nueva visión de la geomorfología de cauces
naturales en Estudios de calidad de aguas. Revista
ACODAL; No.235. (2014):
[24] Chow V.T. Hidráulica de canales abiertos.
McGraw Hill, New York. (2004).
[25] Langbein W. & Leopold L. River meander-
Theory of minimum variance. Env.Sci. (1966).
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
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